Find the arithmetic sequence whose 1st term is 1 and the 1st, 2nd and 6th terms form a geometric sequence
1 / (d + 1) = (d + 1) / (5d + 1)
5d + 1 = (d + 1)^2 ... d^2 - 3d = 0 ... d = 3
An = 3n - 2
To find the arithmetic sequence that satisfies these conditions, we need to find the common difference of the sequence.
Let's assume that the common difference of the arithmetic sequence is d.
Given that the first term is 1, we can write the equation for the first term of the arithmetic sequence as:
a₁ = 1
Now, we are also given that the 1st, 2nd, and 6th terms form a geometric sequence. A geometric sequence is a sequence in which each term is found by multiplying the previous term by a constant ratio.
The ratio of the 2nd term to the 1st term is equal to the ratio of the 6th term to the 2nd term. This gives us:
a₂ / a₁ = a₆ / a₂
Substituting the values we know:
(1 + d) / 1 = a₆ / (1 + d)
Cross-multiplying, we get:
(1 + d)² = a₆
Expanding the square term, we have:
1 + 2d + d² = a₆
Now, we can use the formula for the nth term of an arithmetic sequence to find the 6th term. The formula is:
aₙ = a₁ + (n - 1) × d
Substituting the values we know, we have:
a₆ = 1 + (6 - 1) × d
a₆ = 1 + 5d
Now, we can substitute this expression for a₆ in our previous equation:
1 + 2d + d² = 1 + 5d
Simplifying the equation, we get:
d² - 3d = 0
Factoring out d, we have:
d(d - 3) = 0
This gives us two possible solutions for d:
d = 0 or d = 3
If d = 0, it means that the sequence is constant, which doesn't satisfy the condition of forming a geometric sequence.
Therefore, the common difference of the arithmetic sequence is d = 3.
To find the complete arithmetic sequence, we can plug this value into the formula for the nth term:
aₙ = a₁ + (n - 1) × d
Substituting a₁ = 1 and d = 3, we get:
aₙ = 1 + (n - 1) × 3
Simplifying further, we have:
aₙ = 1 + 3n - 3
aₙ = 3n - 2
So, the arithmetic sequence that satisfies the given conditions is 3n - 2.